Question: Factor the following expression: $5$ $x^2$ $-22$ $x+$ $21$
Answer: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(5)}{(21)} &=& 105 \\ {a} + {b} &=& & & {-22} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $105$ and add them together. The factors that add up to ${-22}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-7}$ and ${b}$ is ${-15}$ $ \begin{eqnarray} {ab} &=& ({-7})({-15}) &=& 105 \\ {a} + {b} &=& {-7} + {-15} &=& -22 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {5}x^2 {-7}x {-15}x +{21} $ Group the terms so that there is a common factor in each group: $ ({5}x^2 {-7}x) + ({-15}x +{21}) $ Factor out the common factors: $ x(5x - 7) - 3(5x - 7) $ Notice how $(5x - 7)$ has become a common factor. Factor this out to find the answer. $(5x - 7)(x - 3)$